A possible cause of the AO polarity reversal from winterto summer in 2010 and its relation to hemispheric extremesummer weather

Yuriko Otomi • Yoshihiro Tachibana •

Tetsu Nakamura

Received: 7 February 2012 / Accepted: 26 April 2012 / Published online: 16 May 2012

� The Author(s) 2012. This article is published with open access at Springerlink.com

Abstract In 2010, the Northern Hemisphere, in particular

Russia and Japan, experienced an abnormally hot summer

characterized by record-breaking warm temperatures and

associated with a strongly positive Arctic Oscillation (AO),

that is, low pressure in the Arctic and high pressure in the

midlatitudes. In contrast, the AO index the previous winter

and spring (2009/2010) was record-breaking negative. The

AO polarity reversal that began in summer 2010 can

explain the abnormally hot summer. The winter sea surface

temperatures (SST) in the North Atlantic Ocean showed a

tripolar anomaly pattern—warm SST anomalies over the

tropics and high latitudes and cold SST anomalies over

the midlatitudes—under the influence of the negative AO.

The warm SST anomalies continued into summer 2010

because of the large oceanic heat capacity. A model sim-

ulation strongly suggested that the AO-related summertime

North Atlantic oceanic warm temperature anomalies

remotely caused blocking highs to form over Europe,

which amplified the positive summertime AO. Thus, a

possible cause of the AO polarity reversal might be the

‘‘memory’’ of the negative winter AO in the North Atlantic

Ocean, suggesting an interseasonal linkage of the AO in

which the oceanic memory of a wintertime negative AO

induces a positive AO in the following summer. Under-

standing of this interseasonal linkage may aid in the long-

term prediction of such abnormal summer events.

Keywords AO � Hot summer 2010 � NAM �Atlantic SST � Blocking

1 Introduction

In Japan, summer 2010 was the warmest in about 100 years

of countrywide measurement records. Moreover, summer

2010 was abnormally hot on a planetary scale. For exam-

ple, Europe, especially Eastern Europe and western Russia,

experienced record-breaking hot temperatures, attributed to

strong atmospheric blocking over the Euro-Russian region

from late June to early August (Matsueda 2011). Addi-

tionally, Barriopedro et al. (2011) showed that the spatial

extent of the record-breaking temperatures of summer 2010

exceeded the area affected by the previous hottest summer

of 2003. Heat anomalies covered almost the entire Eurasian

continent in 2010. In contrast, in winter 2009/2010, just a

half-year earlier, the continent suffered from anomalously

cold weather associated with a record-breaking negative

Arctic Oscillation (AO), which is characterized by positive

sea level pressure anomalies over the Arctic and negative

pressure anomalies over the midlatitudes (Thompson and

Wallace 2000). Moreover, in the same winter, a record-

breaking negative North Atlantic Oscillation (NAO) caused

several severe cold spells over northern and western

Europe (Cattiaux et al. 2010). In fact, the strongest

Y. Otomi � Y. Tachibana (&)

Climate and Ecosystem Dynamics Division,

Mie University, Tsu, Japan

e-mail: tachi@bio.mie-u.ac.jp

Y. Tachibana

Japan Agency for Marine-Earth Science

and Technology, Yokosuka, Japan

T. Nakamura

Center for Global Environmental Research,

National Institute for Environmental Studies,

Tsukuba, Japan

T. Nakamura

Faculty of Environmental Earth Science,

Hokkaido University, Sapporo, Japan

123

Clim Dyn (2013) 40:1939–1947

DOI 10.1007/s00382-012-1386-0

negative AO index of the past 30 years was observed in

December 2009 (Wang and Chen 2010). This drastic

reversal from a record-breaking cold winter to a record-

breaking hot summer is preserved in our memory. What if,

however, that memory could be preserved not only in our

minds but also somewhere on the earth? In particular,

might a memory of the strongly negative wintertime

2009/2010 AO have been preserved in the ocean, because

of its large thermal heat capacity, which could then be

recalled the following summer?

The winter-to-summer evolution of the AO index during

2009/2010 can be summarized as follows: a strongly nega-

tive wintertime AO index continued until May, after which it

abruptly changed, becoming strongly positive in July and

continuing so until the beginning of August. Details of the

AO evolution will be described in the following sections. Ogi

et al. (2005) pointed out that a strongly positive summertime

AO is associated with occurrences of blocking anticyclones,

which contributed to the abnormally hot European summer.

Trigo et al. (2005) also reported that a blocking anticyclone

caused the anomalous hot summer of 2003. The blocking

anticyclone over Europe in summer 2003 was shown to be

part of a planetary-scale wave train, extending from Europe

to eastern Eurasia (Orsolini and Nikulin 2006). The abrupt

change of the AO index from strongly negative to strongly

positive in 2010 thus corresponded to the change from the

abnormally cold winter of 2009/2010 to the abnormally hot

summer of 2010, which shows that the AO index is a good

indicator of abnormal weather on a planetary-scale, and that

extra-seasonal prediction of the AO is a key to long-term

forecasting. In this study, we therefore aimed to examine the

cause of the 2010 change in the AO from strongly negative to

strongly positive.

2 Data and method

The AO was first defined by Thompson and Wallace (2000)

which is based on an invariant empirical orthogonal func-

tion (EOF) spatial pattern throughout the year, and Ogi

et al. (2004) identified seasonal variations of the Northern

Hemisphere annular mode (SV NAM) from 1958 to 2002

by performing an EOF analysis. EOF was applied to a

temporal covariance matrix of geopotential height fields for

individual calendar months using a zonally averaged

monthly geopotential height field from 1,000 to 200 hPa

for the area poleward of 40�N. The daily time series of the

SV NAM index is obtained by projecting daily zonal mean

geopotential height anomalies onto the EOF of each month.

The time series of the SV NAM index shown in Fig. 1 is

calculated by this method.

Ogi et al. (2004) and Tachibana et al. (2010) demon-

strated that in winter, but not in summer, the SV NAM

accords well with the AO defined by Thompson and

Wallace (2000) and used by the Climate Prediction Center

of the U.S. National Oceanic and Atmospheric Adminis-

tration (NOAA/CPC). Ogi et al. (2005) and Tachibana

et al. (2010) also demonstrated that the SV NAM suc-

cessfully captures anomalous summertime weather condi-

tions associated with blocking anticyclones, such as the hot

summer in Europe in 2003, whereas the original AO of

Thompson and Wallace (2000), mainly reflects atmo-

spheric variabilities in winter and cannot capture such a hot

summer. Therefore, Ogi et al. (2005) redefined the sum-

mertime SV NAM as the summer AO. In this study,

therefore, we adopted the SV NAM index defined by Ogi

et al. (2004) as the AO index, and all references to the AO

index in this study mean the SV NAM index.

We used daily data of large-scale atmospheric fields

from the National Centers for Environmental Prediction/

National Center for Atmospheric Research (NCEP/NCAR)

reanalysis data set (Kalnay et al. 1996) to calculate the

Fig. 1 (Top) Time series of the AO index (blue) as defined by Ogi et al.

(2004), who called it the SV NAM index. For reference, the conventional

AO index reported by NOAA/CPC is shown by the gray line. The

vertical axis is dimensionless because the indices are normalized. Tickmarks on the horizontal axis indicate the 1 day of each month. Updated

daily time series from 1958 are available at http://www.bio.

mieu.ac.jp/kankyo/shizen/lab1/AOindex.htm. (Bottom) Time series of

the temperature anomaly (K) at 925 hPa averaged northward of 32.5�N

over the Eurasian continent. Anomalies are calculated according to the

daily climatology of 32 years

1940 Y. Otomi et al.

123

climatology and anomalies of the meteorological field (i.e.,

temperature, geopotential height, and wind velocity).

Monthly means of sea surface temperature (SST) data are

from the NOAA_ERSST_V3 data set, provided by NOAA/

OAR/ESRL PSD (http://www.esrl.noaa.gov/psd/) (Smith

et al. 2008; Xue et al. 2003). We used monthly mean latent

and sensible heat flux data of the Japan 25 year Reanalysis

(JRA-25) and the JMA Climate Data Assimilation System

to examine the atmosphere–ocean interaction (Onogi et al.

2007). Daily and monthly means of outgoing longwave

radiation (OLR) are interpolated OLR data provided by

NOAA/OAR/ESRL PSD (Liebmann and Smith 1996).

Anomaly fields of individual variables are relative to the

multi-year mean climatology from 1979 to 2010 for each

month.

3 Strongly positive AO days

The winter-to-summer evolution of the AO index (Fig. 1)

showed a strongly negative AO in winter 2009/2010 that

lasted through May, followed by an abrupt change to

strongly positive values in July and August 2010. In par-

ticular, the AO index was extremely positive from 10 July

to 4 August 2010, coinciding with a period of abnormally

hot days in eastern Europe and the Russian far east.

Moreover, the AO index in winter and summer accords

well with changes in the temperature anomaly for the

Eurasian continent over the same period (Fig. 1, lower

panel), although the AO index in spring did not accord well

with the temperature. Time-mean atmospheric fields during

the strongly positive AO period are shown in Fig. 2. The

temperature anomaly field at 850 hPa shows two obvious

exceptionally hot areas, one centered over eastern Europe

and the other over the Russian far east. Between these two

hot areas, cold anomaly areas can be seen over central

Siberia and the Arctic. At 300 hPa, a negative geopotential

height anomaly is seen over the Arctic region that elon-

gates southward toward central Siberia, whereas positive

anomalies characterize the midlatitudes of the Northern

Hemisphere. Over eastern Europe, Mongolia, the Russian

far east, and the eastern North Pacific Ocean the positive

anomalies are particularly strong. This pattern is very

similar to the positive summer AO pattern observed during

the unusually hot summer of 2003 (Ogi et al. 2005). In

summer 2010, the geopotential height contours meandered

widely around the Arctic region, indicating that the polar

jet stream meandered similarly. In addition, the jet stream

split into north and south branches over eastern Europe and

the Russian far east, suggesting the existence of a blocking

high. At 300 hPa, wave-activity fluxes (Fig. 2a, green

arrows) over the polar jet were oriented from Europe to

south of Alaska along the longitudinal circle, and they were

particularly strong over eastern Europe and the Russian far

east, suggesting Rossby wave sources in those areas. The

existence of a double jet stream structure is also apparent in

the two zonal wind maxima seen at about 72�N and 45�N

along 135�E (Fig. 2c). From the surface to the upper tro-

posphere at about 55�N, where the largest negative wind

anomaly is observed, the wind direction is easterly. This

large-scale pattern in 2010 is consistent with the findings of

Ogi et al. (2004), who reported an enhanced double jet in

the positive phase of the summer AO.

4 Oceanic footprint left by the previous winter’s

negative AO

In the North Atlantic Ocean, a tripolar SST anomaly pat-

tern, warm in the high latitudes, cool in the midlatitudes,

and warm in the tropics, persisted from January to August

2010 (Fig. 3). This tripolar pattern is typical of a negative

wintertime NAO (e.g., Rodwell et al. 1999; Tanimoto and

Xie 2002). In fact, the geopotential height anomaly field at

500 hPa in winter (DJF) 2009/2010 showed the typical

pattern for the negative phase of the NAO (Fig. 4). The

strong negative phase of the AO index in the winter of

2009/2010 corresponded to the negative phase of the NAO

(Fig. 4a, b). The temperature anomaly at 850 hPa of winter

in the region of high-latitude and mid-latitude North

Atlantic corresponded well to the total latent and sensible

heat flux anomaly in January and February (Figs. 3, 4c).

Similar to the tripolar SST anomaly pattern, the total latent

and sensible heat flux anomaly in January and February

was also tripolar (Fig. 3): a downward flux anomaly

occurred over high latitudes and the tropical North Atlan-

tic, and an upward flux anomaly was observed over the

midlatitudes. The downward anomaly in the high latitudes

and tropical North Atlantic lasted until April, but the sign

of the latent and sensible heat flux anomaly reversed from

downward to upward over the tropical North Atlantic in

May and June and over the North Atlantic high latitudes in

July and August, whereas the warm SST anomaly in the

high latitudes and tropical North Atlantic continued into

the summer. The monthly mean tropical North Atlantic

SST from January to August was the warmest observed in

the 32 years from 1979 to 2010. On the strongly positive

AO days, the OLR anomaly over the North Atlantic was

strongly negative over the Caribbean Sea (Fig. 5). The

negative OLR anomaly area, which was characterized by

strong convective activity, roughly coincided with the area

of the warm SST and upward sensible and latent heat flux

anomalies in summer. In addition, the wind field anomaly

in the lower troposphere was cyclonic in the central area

of the negative OLR anomaly in the tropical North

Atlantic.

A possible cause of the AO polarity 1941

123

5 Steady responses to the oceanic forcing

in the Atlantic region

The results presented in Sects. 3 and 4 suggest that the SST

anomaly pattern in the Atlantic Ocean in summer 2010

remotely influenced the midlatitude atmospheric circula-

tion. Many studies have investigated North Atlantic influ-

ences on the midlatitudes. For example, Cassou et al.

(2005) showed that atmospheric convection over the trop-

ical Atlantic leads to an anticyclonic anomaly over Europe.

To make a robust assessment of the SST influence, a sen-

sitivity experiment conducted with a numerical model that

simulated the atmospheric responses to a given anomalous

SST in the Atlantic region would be useful. However, it is

generally difficult to simulate the evolution of a blocking

high with an atmospheric general circulation model

because of the strong non-linearity of blocking highs. Here

we adopted instead a simple linear model, formulated by

Watanabe and Kimoto (2000), to simulate the atmospheric

response to the ocean. In this model, a spectral primitive

equation is linearized about the climatological mean state.

We used a version with T42L20 resolution. A steady

response X with the basic state �X is derived by using an

equation with the matrix form.

Lð �XÞX ¼ Q;

where Q is the temperature forcing vector and L is the

linear dynamical operator (for details, see ‘‘Appendix 1’’

Fig. 2 a Time-mean

geopotential height at 300 hPa,

b temperature at 850 hPa

(T850), and c vertical crosssection of the eastward wind

component at 135�E in the

Northern Hemisphere during

strongly positive AO days from

10 July to 4 August 2010.

Contours show time-mean

values of geopotential height

(a, contour interval 100 m),

temperature (b, contour interval

5 K), and wind speed (c,

contour interval 5 m s-1), and

the color shading shows (a) the

geopotential height anomaly,

(b) the temperature anomaly,

and (c) the zonal wind anomaly

from climatological temporal

means. The green arrows in

(a) show the wave activity flux

(m2 s-2) at 300 hPa as

formulated by Takaya and

Nakamura (2001), with the scale

shown by the arrow in the upperright corner

1942 Y. Otomi et al.

123

and Watanabe and Kimoto 2000). We defined �X as the

climatological mean in July, derived from the monthly

mean NCEP/NCAR reanalysis data set, and Q as the

diabatic heating anomalies in July 2010, obtained by a

conventional Q1 analysis (Yanai et al. 1973, see ‘‘Appendix

2’’ for details) using the 6 h NCEP/NCAR reanalysis data.

A model simulation is useful to examine whether the

oceanic memory in the Atlantic region is essential to the

generation of the blocking high. Thus, we separately cal-

culated Q at levels from 0.7 to 0.3 r (r-coordinate system)

for the Atlantic Ocean region from 90�W to 30�E (Fig. 6a)

and the Eurasian and African continental region from 0�E to

150�E (Fig. 6b) in the Northern Hemisphere. The horizontal

distribution of the diabatic heating anomalies in July 2010

(Fig. 6a) is acceptably similar to the SST tripolar pattern

(Fig. 3, JA, bottom left). In particular, warmer SST regions

(around the British Isles and near the equator) correspond to

positive anomalies in the temperature forcings. The coin-

cidence of the regions of warmer SST and positive heating

anomalies indicates that the Atlantic Ocean heats the

Fig. 3 Evolution of (left column) SST and its anomaly, and (rightcolumn) the sum of the latent and sensible heat fluxes and its

anomaly, from January to August 2010. Contours show 2 month

mean values, and the color shading shows the anomalies (deviations

from the climatological temporal mean). JF, January and February;

MA, March and April; MJ, May and June; JA, July and August. The

contour interval for SST is 3 �C, and that for the flux is 40 Wm-2.

Here, a positive flux (i.e., upward flux) is defined as from the ocean to

the atmosphere. Red or blue shading in the right panels thus indicates

anomalous heating or cooling of the ocean, respectively

A possible cause of the AO polarity 1943

123

atmosphere in those regions. On the other hand, in the

continental region, although heating anomalies are signifi-

cant over low-latitude Africa where OLR anomalies were

negative (Fig. 5), cooling anomalies are dominant in wes-

tern Russia and northern Europe but with except of

Scandinavia.

Figure 7 shows the steady responses of zonal wind and

geopotential height at 300 hPa to the given Q. In the

responses to the forcing of the Atlantic Ocean, an anticy-

clonic height anomaly with a maximum amplitude

exceeding 30 m is obvious over northern Europe (Fig. 7c),

and the corresponding zonal wind anomaly strengthens the

climatological double-jet structure (Fig. 7a). On the other

hand, in the responses to the forcing of the Eurasian and

African continent, a strong cyclonic anomaly is seen over

northern Europe/western Russia (Fig. 7d), which may be a

counter response to dynamical (i.e., adiabatic) heating due

to the blocking high developed there. Thus, our model

simulation strongly suggests that atmospheric heating

related to the tripolar pattern of the Atlantic SST anomaly

is one of the main causes of the blocking high over Europe.

6 Discussion

Taking together the results presented in Sects. 3, 4, and 5,

we suggest that an oceanic memory of the strongly nega-

tive wintertime AO may have influenced the strongly

positive summertime AO. A negative wintertime NAO

would cause warm SST anomalies in high- and low-lati-

tude regions of the Atlantic, as suggested by Xie and

Tanimoto (1998) and Tanimoto and Xie (2002). Because

the horizontal structures of the NAO and the AO in the

Atlantic sector in winter 2009/2010 are similar (See

Fig. 4), the strongly negative wintertime AO would

maintain the warm SST anomaly in this region. The

Fig. 4 Winter 2009/2010

(December, January, and

February) mean geopotential

height at 1000 hPa (a) and

500 hPa (b) and temperature

at 850 hPa (c). Contours show

winter mean values of

geopotential height (contour

interval 50 m) and temperature

(contour interval 5 K). The

color shading shows the

geopotential height anomaly or

the temperature anomaly from

climatological temporal means

Fig. 5 OLR anomaly (color scale, Wm-2), defined as the deviation

from the climatological temporal mean, on strongly positive AO days.

Arrows show the surface wind anomaly (m s-1) on strongly positive

AO days, with the scale shown by the arrow below the lower rightcorner

Fig. 6 Vertically averaged (from 0.7 to 0.3 r) diabatic heating

derived from a conventional Q1 analysis (Yanai et al. 1973). The

color scale indicates the heating anomaly (K day-1) in July 2010

(deviation from the July climatological mean). a Heating anomalies

over only the Atlantic Ocean (from 90�W to 30�E, from 0�N to 90�N).

b Heating anomalies over only the Eurasian and African continent

(from 0�E to 150�E, from 0�N to 90�N)

1944 Y. Otomi et al.

123

downward latent and sensible heat flux anomaly over the

high latitudes and the tropical Atlantic (Fig. 3) in winter

and spring indicates that anomalous heating of the ocean by

the atmosphere occurred from winter to spring during the

strongly negative phase of the AO in winter 2009/2010.

Because the thermal heat capacity of the ocean is large, the

sea surface stored this warmth (i.e., the SST anomaly

remained positive) into the following summer.

In May and June, the heat flux anomaly changed from

downward to upward in the tropics (see Fig. 3), and in July

and August, the center of the upward anomaly moved

westward. The area of the upward heat flux anomaly

coincided with the area of the warm SST anomaly from

May to August. The warm SST during the summer fol-

lowing the strongly negative wintertime AO therefore

heated the atmosphere, activating atmospheric convection.

The OLR anomalies also indicate high convective activity

in the tropical Atlantic region (Fig. 5), suggesting a remote

influence of the Atlantic SST upon the occurrence of an

anticyclone over Europe. This Atlantic SST influence has

been pointed out by many studies (e.g., Cassou et al. 2005;

Garcıa-Serrano et al. 2008). Garcıa-Serrano et al. (2008)

showed that a midlatitude anticyclonic anomaly related to

tropical convection can excite a Rossby wave. Our

numerical experiment using the linear model showed that

the atmospheric response to the tripolar SST pattern clearly

resulted in an anomalous height and wind pattern that

caused a blocking high over Europe (Figs. 6, 7), however,

the modeled geopotential amplitude is weaker than the

observations. This discrepancy is because a linear model

cannot represent the dynamical instability due to, for

example, wave–wave interaction. Therefore, the model

indicates that although the oceanic memory in the Atlantic

is a trigger, by itself it is insufficient to cause a blocking

high to develop. Weak, positive OLR anomalies along the

Gulf Stream were associated with anticyclonic surface

winds on strongly positive AO days (Fig. 5). The observed

wave activity flux (Fig. 2a) also seems to emanate from

that region. This midlatitude signature implies that

strengthening of the positive geopotential anomalies over

Europe was associated with the Atlantic tripolar SST

anomaly.

Fig. 7 a Steady responses of

zonal wind at 300 hPa (U300)

to the heating anomalies over

the Atlantic Ocean,

corresponding to those shown in

Fig. 6a. Red/blue shading

indicates positive/negative

anomalies with units (m s-1).

b As in (a) except for responses

to the heating anomalies over

the Eurasian and African

continent, corresponding to

those shown in Fig. 6b. (c and

d) As in (a) and (b) respectively

except for geopotential height at

300 hPa (Z300) with units (m)

A possible cause of the AO polarity 1945

123

The positive geopotential anomaly in the area of the

polar jet stream caused eastward propagation of Rossby

waves, and the unusual amplification of Rossby waves

might have led to the formation of blocking anticyclones.

These findings are in agreement with previous studies. For

example, Tachibana et al. (2010) reported that a blocking

anticyclone over the Atlantic sector that induces blocking

over the Russian Far East is associated with a long-lasting,

strongly positive AO caused by wave–mean flow interac-

tions. As a result of these interactions, the positive AO

pressure pattern can continue for a long time. In addition,

Orsolini and Nikulin (2006) pointed out that the blocking

anticyclone over Europe in summer 2003 was part of a

wave train extending from Europe to eastern Eurasia. The

linear model did not simulate an anticyclonic anomaly in

the Russian Far East. To simulate the influence of an

anomalous wintertime negative AO on an anomalous

positive AO in the following summer due to a long-lasting

oceanic memory, an atmosphere–ocean coupled high-res-

olution model simulation is needed. We reserve this

experiment for future studies.

Of course, the set of processes introduced here is just one

possible explanation for the formation of the strongly positive

summer AO in 2010. For example, summertime SST anom-

alies in the Mediterranean Sea (Feudale and Shukla 2010)

might simultaneously induce a strongly positive summer AO.

Although the effect of the oceanic memory of a negative AO

during the previous winter might be smaller than the effects of

simultaneous events, the previous winter’s footprint may at

least play a role in the reversal of the AO polarity from a

strongly negative wintertime AO to a strongly positive

summertime AO. If this reversal pattern recurs, it might be

possible to predict the summer AO from the wintertime AO.

The more negative the winter AO anomaly is, the deeper the

footprint left in the ocean would be, suggesting that a winter-

to-summer reversal of the AO might occur only in years when

the negative wintertime AO anomaly is large. In addition to

an oceanic memory effect, other memory effects such as

anomalous snow accumulation on the Eurasian continent or

elsewhere in the Northern Hemisphere, as suggested by Ogi

et al. (2003) and Barriopedro et al. (2006), may also con-

tribute to the reversal of AO polarity. To test these possibil-

ities, statistical analyses of multi-year data and simulation by

a full coupled atmosphere–ocean–land global climate model

are the next step.

Acknowledgments We extend special thanks to V. A. Alexeef,

M. Honda, H. E. Hori, and J. Inoue for their very helpful comments on

this study. We also thank two anonymous reviewers for their valuable

comments and suggestions to improve the quality of the paper. This

study was supported by Grant-in-Aid for challenging Exploratory

Research 22654055, and a part of this study was supported by ‘‘Green

Network of Excellence’’ Program (GRENE Program) Arctic Climate

Change Research Project.

Open Access This article is distributed under the terms of the

Creative Commons Attribution License which permits any use, dis-

tribution, and reproduction in any medium, provided the original

author(s) and the source are credited.

Appendix 1: A brief description of the linear baroclinic

model

In this study, we used a linear baroclinic model (LBM)

identical to one used by Watanabe and Kimoto (2000).

They exactly linearized primitive equations in which the

prognostic variables are vorticity (f), divergence (D),

temperature (T), and surface pressure (p ¼ ln Ps). Using

a state vector X f;D; T; pð Þ, a dynamical system can be

represented as.

dtX þ Lþ NLð ÞX ¼ F; ð1Þ

where L and NL are the linear and nonlinear parts of a

dynamical operator that consists of, for example,

advection, Coriolis, pressure gradient, and dissipation

terms. F is a forcing (in this study, we used diabatic

heating Q). Now, a state vector X can be decomposed into

a basic state �X and a perturbation part X0, i.e., X ¼ �X þ X0.

Equation (1) is linearized about a basic state �X. Then we

consider the steady problem, neglecting the nonlinear part,

obtaining a set of linear equations for the perturbation of

the prognostic variables X0:

LX0 ¼ F0: ð2Þ

Note that a linear dynamical operator L is now a function of

the basic state, i.e., L � Lð �XÞ, obtained following Hoskins

and Karoly (1981). Equation (2) can be solved by using an

inverse matrix of L:

X0 ¼ L�1F0: ð3Þ

Equation (3) gives us a steady response to a given

forcing. Linearized equations are not necessarily required

to obtain a steady response. For a process such as the

development of a blocking high, in which internal

instability is important, linear responses may be weaker

than expected. However, LBM is useful for diagnosing the

primary response of the atmosphere without secondary

feedback due to, for example, changes in heat fluxes from

surfaces.

Appendix 2: Estimation of diabatic heating

in the atmosphere

Atmospheric diabatic heating can be estimated as a residual

term from a heat budget analysis of the thermodynamical

1946 Y. Otomi et al.

123

equation. Here, diabatic heating Q is defined as follows:

Q � Cp p=p0ð ÞR

Cp ohot þ v � rhþ x oh

op

� �, where Cp is specific

heat of dry air at constant pressure, R gas constant for

dry air, p pressure, p0 standard sea level pressure (=

1,000 hPa), h potential temperature, v horizontal wind

vector, and x pressure velocity. Q includes not only sen-

sible heat but also latent and radiative heat. In general, in

the free troposphere, the latent heat associated with the

condensation and the evaporation of water vapor is domi-

nant, although the radiative heating may be large in a

specific situation such as at the cloud-top.

This estimation method was first introduced by Yanai

et al. (1973). They used Q (named Q1 in their study) as an

apparent heat source and further defined Q2, which is a

residual term of the water vapor budget equation, as an

apparent moisture (i.e., latent heat) sink to estimate prop-

erties of the tropical cloud cluster from the observed large-

scale heat and moisture budget. They also pointed out that

this method is useful for determining how a large-scale air

is heated diabatically, including both latent and radiative

heating.

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A possible cause of the AO polarity 1947

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